First Elements of Thermal Neutron Scattering Theory (II) Daniele Colognesi Istituto dei Sistemi Complessi, Consiglio Nazionale delle Ricerche, Sesto Fiorentino (FI) - Italy

Talk outlines

4) Inelastic scattering from fluids (intro) Disordered systems (gasses, liquids, glasses, amorphous solids etc.): atomic order only at short range (if existing). For simplicity’s sake only monatomic fluid systems are considered here. key quantities: density, , constant, and pair correlation function, g(r) connected to the static structure factor, S(Q), via a 3D spatial Fourier transform:

where both S(Q) and g(r) exhibit some special values at their extremes: Since S(Q)=I(Q,t=0), it is possible to generalize g(r) by introducing the time-dependent pair correlation function, G(r,t):

and the time-dependent self pair correlation function, G self (r,t):

where the t=0 values of G(r,t) and G self (r,t) are: No elastic scattering,  (  ), in fluids! the elastic components in S(Q,  ) and S self (Q,  ) come from the asymptotic values of I(Q,t) and I self (Q,t):

Due to the asymptotic loss of time correlation, and making use of  =  i  (r-r i ) , one writes: so, finally:

Gas of non-interacting distinguishable particles: a useful “toy model”. No particle correlation: S(Q,  )  S self (Q,  ). Starting from the definitions: one writes:

After some simple algebra: Very important for epithermal neutron scattering! recoil Doppler broadening

Coherent inelastic scattering from liquids a.k.a. “Neutron Brillouin Scattering”: the acoustic phonons become pseudo-phonons (damped, dispersed). A new undispersed excitation appears too. Very complex, not discussed here.

Liquid Al g(r) Liquid Ni S(Q)

Incoherent inelastic scattering from liquids: the elastic component becomes quasi-elastic (diffusive motions), not discussed here in great detail. On the contrary, the inelastic component is not too dissimilar from the crystal case (pseudo- phononic excitations).

Starting from the well-known: it is possible to show (Rahman, 1962) that:

where we made use of the Gaussian approximation in Q. The t-dependent factor has apparently a tough aspect: but it is actually equal to Q -2 [B(Q,0)-B(Q,t)]. Then f liq (  ) has to be analogous to g(  ) in solids… Surprising! Let’s study it, starting from the velocity self- correlation function of an atom in a crystal: c vv (t).

Expanding in normal modes through the Bloch theorem, one gets (in the isotropic case): It applies to f liq (  ) too. Using the fluctuation- dissipation theorem, linking Re[c vv (t)] with Im[c vv (t)], one writes:

However, there is a property distinguishing f liq (  ) from g(  ): where D is the self-diffusion coefficient, while g(0)=0.

Example: liquid para- hydrogen, measured on TOSCA at T=14.3 K (Celli et al. 2002) and simulated through Centroid Monte Carlo Dynamics (Kinugawa, 1998).

5) Vibrational spectroscopy from molecules chemical-physical spectroscopy: studying the forces that: -bind the atoms in a molecule [covalent bond: E  400 KJ/mol]. -keep the functional groups close to one another [hydrogen bond: E  20 KJ/mol]. -place the molecules according to a certain order in a crystalline lattice [molec. crystals: E  2 KJ/mol]. Wide range of energies! Here only intra-molecular modes (vibrational spectroscopy).

Cross-section summary H case (ideal incoherent scatterer):  inc =80.27 b,  coh =1.76 b  Proton selection rule D case (quite different):  inc =2.05 b,  coh =5.59 b Then only incoherent scattering will be considered in the rest of this talk!

Comparing various spectroscopies  (neutron)  m 2 /molec.  (Raman)  m 2 /molec.  (IR)  m 2 /molec. Why neutron spectroscopy ? 1.In Raman polarizability generally grows along with Z: possible problems in detecting H. 2.In IR (sensitive to the electric dipole) the H-bond gives rise to a large signal, but it is distorted by the so-called electric anharmonicity (not vibrational). 3.Molecules with elevate symmetry: many modes are optically inactive (e.g. in C 60 up to 70%!).

4. Direct relationship between neutron spectra and vibrational eigenvectors. Conclusions Neutron spectroscopy is complementary to optical spectroscopies (Raman and IR) and is often essential for studying proton dynamics! Example: nadic anhydride (C 9 H 8 O 3 ) on TOSCA

Molecular vibrations and normal modes Polyatomic Molecules: N atoms instantaneously in the positions {r α }, vibrating around their equilibrium positions {r α0 }: r α = r α0 +u α Normal modes 3 traslations 3 rotations (2 if linear) 3N-6 vibrations (3N-5 if linear) Translations elimination (center-of-mass fixed):  α m α r α =  α m α r α0 =R   α m α u α =0

Rotations elimination (small oscillations):  α m α r α  v α = J=0  α m α r α0  t u α   α m α r α0  u α =cost.  0 The normal modes of a molecule can be classified according to the character of the atomic motions, starting from the symmetry of the equilibrium configuration of the molecule (group theory). General Theory of normal modes with s d.o.f. q i : u i =q i -q i0

One gets s Lagrange equations:  Oscillating test solutions:  Characteristic equation : (in general one has s real and positive roots:  1,…  s )  Eigenvectors a j (s) :

General solution: Example: normal modes in H 2 O a. Symmetric stretching b. Bending c. Anti-symmetric stretching

Normal mode quantization

Diffusion from a harmonic oscillator The mono-dimensional harmonic oscillator is then the simplified prototype of the true intra-molecular vibrations: ~1000 cm -1 <   0 <4400 cm -1 (H-H):

Typical experiment : T=20 K (i.e. 14 cm -1 <<   0 ) then: from which: where  u 2  0 is the mean square displacement (at T=0).

Again on the harmonic oscillator Mass problem: what is μ in a molecule? It depends on all the atomic masses, but M H obviously plays a primary role! However, in general, μ  M H. Elastic Line: there is no exchange of energy between oscillator and neutron, then n=0. It is intense, but it decreases rapidly with Q. Then it will be neglected:

Fundamental: for n=1 there is a peak centered at  0, while in Q one gets a competition between the Debye-Waller factor and the term Q 2  u 2  0 : The maximum of S n=1 (Q,E) appears at Q 2 =  u 2  0. So, the ideal measurement conditions for H are: k 1 <<k 0  k 0  Q for any value of E. Namely:

Overtones: excitations from the ground state (n=0) to states higher than the first (i.e. n=2,3…): considering that: one obtains:

The relative intensity of the overtones (with respect to n=1) quickly decreases along with μ. It is important to separate the high-frequency fundamental excitations from the overtones. Example: fundamental and overtones in ZrH 2, almost a harmonic oscillator (three- dimensional).

Anharmonicy Ideal vibrational model: set of decoupled harmonic oscillators (normal modes). Anharmonicity: breaking of the harmonic approximation, implying inseparability and mixing of normal modes. In practice overtones are not simple multiples of the fundamental frequency any more, i.e. there is an anharmonicity constant, . One often has that  >0 (e.g. in the Morse potential).

In practice, in real molecules one uses a pseudo- harmonic approach in which the structure factor for a single atomic species is approximated by: where n labels the sum over the overtones and k the multi-convolution in E over the normal modes, from which:

6) Incoherent inelastic scattering from molecular crystals External molecular modes So far only isolated molecules have been dealt with, having a fixed center-of-mass (no recoil). In reality, at low temperature, one observes molecular crystals kept together by inter- molecular interations: weak (van der Waals), medium (H bond), or strong (covalent). External modes (p k, lattice vibrations and undistorted librations): in general (but not always…) softer than the internal ones (e.g. lattice v. ~150 cm -1 ).

Similarly to what seen for the internal modes, an external structure factorfor the molecular lattice can be defined: Similarly to what seen for the internal modes, an external structure factor for the molecular lattice can be defined: making implicitly use of the decoupling hypothesis between internal and external modes:

using the distributive property of the convolution one gets: then for each internal mode  k there is also a shifted replica of all the external spectrum {p k’ } (phononic branch), but with a strong intensity reduction due to the external Debye-Waller factor:

At low Q, S orig (Q,E) is intense and S bran (Q,E) has a shape similar to that of S ext (Q,E) (but translated). At high Q, S bran (Q,E) is dominated by the multiphonon terms (difficult to be simulated). Comparison to the mean square displacements worked out by diffraction: Discrepancies between B iso and the inelastic mean square displacements: static disorder

Example: hexamethylenetetramine (C 6 H 12 N 4 ) on TOSCA

Anisotropy and spherical mean We have seen that, owing to the presence of various normal modes, scattering depends on the orientation of Q with respect to the molecule (anisotropy). Toy-model: 1-D harmonic oscillators with frequency  x, all oriented along the x axis(e.g. parallel diatomic molecules and one lattice site only):

S n=1 (Q,E) is maximum for φ=0 (Q||x) and zero for φ=90 o (Q  x). Similar to E in IR. It is also defined a displacement tensor B ij : In practice the powder spectrum will be a spherical average containing various modes  i :

One can prove that a good approximation of the spherical mean is given, for the fundamental, by: where: This expression is formally identical to the isotropic harmonic oscillator one: all the vibrations are visible, but wakened by a factor 1/3.

 Example of the anisotropy importance in highly-oriented (>90%) polyethylene  –––––––––– c ––––––– 

Q  c (calc. by Lynch et al.) Q||c (calc. by Lynch et al.)  Example: lattice modes in highly-oriented polyethylene simulated for TOSCA

7) Some applications to soft matter What is soft matter? Soft matter: it is often macroscopically and mechanically soft, either as a melt or in solution. On a short scale: there is a mesoscopic order together with weak intermolecular force constants [   v /(3k B T)  1]. It is in between solids and liquids (both for its structure and for its dynamics). It is not yet rigorously defined. Main classes (after Hamley, 1999): polymers, colloids, amphiphiles and liquid crystals. Good picture, but there is still some overlap!

What is spectroscopy? A microscopic dynamical technique: spectral analysis (k,  ) of a probe, before and after its interaction with a sample. Absorption (  0 ) or scattering (  k,  ). Basic idea:  0  2  /t;  |k|  2  /|r| and  2  /t. Differences: i) probe [e.m. waves:  =c|k|, neutrons:  =  |k| 2 /(2m n )]; ii) interaction [e.m. waves:  A  j  , neutrons: (2   2 /m n ) b  (r)].

Main spectroscopic techniques for soft matter i) Nuclear Magnetic Resonance (NMR). ii) Infrared absorption and Raman scattering (IR and Raman). iii) Dielectric Spectroscopy iv) Visible and ultraviolet optical spectroscopy v) Inelastic neutron scattering (INS). E = E i – E f Q = k i – k f

Why INS for soft matter? Limitations of IR and Raman: selection rules (from  f|D|i  and  f|P|i  ). Group theory. General problems with optical techniques: i) dispersion and acoustic modes; ii) selection rules; iii) proton visibility; iv) spectral interpretation. INS is always complementary and often essential

i) Dispersion and acoustic modes collective modes dispersion:  =  j (q), con 0<|q|<2  /a  20 nm -1. What |q| can be obtained through e.m. waves? Green light (E=2.41 eV): |q|= nm -1  0… X-rays are needed (E>1 KeV): IXS. Acoustic modes:  ac (|q|  0)=c s |q|  0. Thermal neutrons: (E=25.85 meV): |q|=35.2 nm -1.

ii-iii) Selection rules and proton visibility High symmetry: many modes are optically inactive (C 60 : 70%!). Neutrons: pseudo-selection rule for H (  H =81.67 barn >>  x  1-8 barn). Isotopic substitution: H  D (  D =7.63 barn). Proton visibility in Raman: Tr (P) grows along with Z. Proton visibility in IR: strong signal for H-bonds (e.g. O-H), but there is also the electric anharmonicity (distortions). iv) Spectral Interpretation Direct interpretation of the spectral line intensities: vibrational eigenvectors (IR and Raman:  f|D|i ,  f|P|i  ). Example: one-dimensional harmonic oscillator (at T=0):

Example: isotopic substitution in potassium hydrogen phthalate. Two hydrogen-bond modes are clearly pointed out.

Would you like to know more? (from easy to difficult) “Introduction to the Theory of Thermal Neutron Scattering” by G. L. Squires (1978). “Vibrational Spectroscopy with Neutrons” by P. C. H. Mitchell et al. (2005). “Molecular Spectroscopy with Neutrons” by H. Boutin and S. Yip (1968). “Neutron Scattering in Condensed Matter Physics” by A. Furrer, J. Mesot and T. Straessle (2009). “Slow Neutrons” by V. F. Turchin (1965). “Theory of Neutron Scattering from Condensed Matter I” by S. W. Lovesey (1984).

Acknowledgements Many thanks to: Dr. R. Senesi (Univ. Roma II) for the kind invitation to talk. The audience for its attention and interest.